# Tag Archives: word algebra

maybe, for the presentation $\langle a,b,c\mid a^2=1, b^2=1 , c^2=1\rangle$, is this the its Cayley’s graph? 1 Comment

## integers assigned respectively integers assigned respectively

Let me tell that these are words for the group presented as $\langle a,\ b\ \mid \ a^2=e, \ b^2=e\rangle$

Este grupo tiene una infinidad de elementos de orden dos: todas las palabras de longitud impar como $w=ababa$ cuando se hace $w^2=(ababa)(ababa)=1$.

There are  words as $w_1=ababab$ that with the word $w_2=bababa$ they do $w_1w_2=w_2w_1=1$

Type $W_1$ words are like $w=(ab)^n$ and $W_2$ and words as $w=(ba)^m$, then they form a subgroup $H$ which is isomorphic to $\mathbb{Z}$. This via $(ab)^n\mapsto n, (ba)^n\mapsto -n$.

Este subgrupo arma solo dos clases laterales ${\mathbb{Z}}_2*{\mathbb{Z}}_2/H=\{eH,\ aH=bH\}$. Entonces vemos que $1\to{\mathbb{Z}}\to{\mathbb{Z}}_2*{\mathbb{Z}}_2\to{\mathbb{Z}}_2\to1$

es exacta.

That is, ${\mathbb{Z}}_2*{\mathbb{Z}}$ is  an extension of $\mathbb{Z}$ by ${\mathbb{Z}}_2$.

Otra posible extensión es la trivial: $1\to{\mathbb{Z}}\to{\mathbb{Z}}\oplus{\mathbb{Z}}_2\to{\mathbb{Z}}_2\to1$

It is known that all extensions $E$ in  an exact sequence $1\to{\mathbb{Z}}\to E\to{\mathbb{Z}}_2\to1$ are in bijection within the morphisms ${\mathbb{Z}}_2\to{\rm out}\mathbb{Z}\cong{\mathbb{Z}}_2$

1 Comment

Filed under algebra, group theory, math

## paraphrasis 1

tensors are like any other machines,

they are either a benefit or a hazard,

if they’re a benefit, it’s not my problem…

: |